Motion of a rigid disc in an elastic solid

1971 ◽  
Vol 61 (6) ◽  
pp. 1717-1729
Author(s):  
A. K. Mal
Keyword(s):  
Exact Solution ◽  
Elastic Solid ◽  
Approximate Solutions ◽  
Compressional Wave ◽  
Fredholm Integral Equations ◽  
High Frequencies ◽  
Incident Plane ◽  
Smooth Contact ◽  
Rigid Disc ◽  
Quantities Of Interest

abstract A rigid circular disc embedded in an infinite, isotropic elastic solid is assumed to be excited by a normally-incident plane, harmonic compressional wave. The disc is assumed to be either in perfectly welded or perfectly smooth contact with the surrounding solid. In each case, the problem is reduced to the solution of Fredholm integral equations of the second kind. For the case of the welded contact, exact numeral solutions valid at any given frequency are given. Approximate solutions valid at low and high frequencies are obtained for both cases and are compared with the exact solution. The displacement and compliance of the disc as well as other quantities of interest are presented graphically as functions of frequency.

scholarly journals Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations

2020 ◽  
Vol 17 (3(Suppl.)) ◽  
pp. 1010
Author(s):  
Pakhshan M. Hasan ◽  
Nejmaddin Abdulla Sulaiman
Keyword(s):  
Exact Solution ◽  
Linear System ◽  
Perturbation Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Homotopy Perturbation

           In this paper, the homotopy perturbation method (HPM) is presented for treating a linear system of second-kind mixed Volterra-Fredholm integral equations. The method is based on constructing the series whose summation is the solution of the considered system. Convergence of constructed series is discussed and its proof is given; also, the error estimation is obtained. Algorithm is suggested and applied on several examples and the results are computed by using MATLAB (R2015a). To show the accuracy of the results and the effectiveness of the method, the approximate solutions of some examples are compared with the exact solution by computing the absolute errors.

Wave Function Expansions and Perturbation Method for the Diffraction of Elastic Waves by a Parabolic Cylinder

1967 ◽  
Vol 34 (4) ◽  
pp. 915-920 ◽  
Author(s):  
S. A. Thau ◽  
Yih-Hsing Pao
Keyword(s):  
Wave Function ◽  
Separation Of Variables ◽  
Low Frequency ◽  
Elastic Solid ◽  
Compressional Wave ◽  
Parabolic Cylinder ◽  
Infinite Matrix ◽  
Elastic Solids ◽  
Incident Plane ◽  
Scattered Fields

The dynamic response, including the stresses at the surface, of a rigid parabolic cylinder in an infinite elastic solid is studied for an incident plane compressional wave. The method of separation of variables in parabolic coordinates is used. With the wave function for one of the scattered waves expanded into a series of those for the other wave, the total scattered fields are then determined numerically by inverting a truncated infinite matrix. The same problem is solved also by a recently developed method of perturbation which describes the two waves in elastic solids in terms of wave functions with a common wave speed. With the latter method, the total scattered waves are determined analytically for the various orders of perturbation, and these results supplement the numerical wave function expansion results in the low-frequency range.

scholarly journals An Adaptive Approach to Solutions of Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Nebiye Korkmaz ◽  
Zekeriya Güney
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Iteration Method ◽  
Legendre Polynomials ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Coarse Mesh ◽  
Fine Mesh

As an approach to approximate solutions of Fredholm integral equations of the second kind, adaptive hp-refinement is used firstly together with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution. The linear hat functions and modified integrated Legendre polynomials are used as basis functions for the approximations. The most appropriate refinement is determined by an optimization problem given by Demkowicz, 2007. During the calculationsL2-projections of approximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated. Depending on the error values, these procedures could be repeated consecutively or different meshes could be used in order to decrease the error values.

scholarly journals A perturbed algorithm for strongly nonlinear variational-like inclusions

2000 ◽  
Vol 62 (3) ◽  
pp. 417-426 ◽  
Author(s):  
C.-H. Lee ◽  
Q. H. Ansari ◽  
J.-C. Yao
Keyword(s):  
Exact Solution ◽  
General Setting ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Special Cases ◽  
The One

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

scholarly journals Satisfying positivity requirement in the Beyond Complex Langevin approach

2018 ◽  
Vol 175 ◽  
pp. 11026 ◽  
Author(s):  
Adam Wyrzykowski ◽  
Błażej Ruba Ruba
Keyword(s):  
Exact Solution ◽  
Approximate Solutions ◽  
Groebner Basis ◽  
Matching Conditions ◽  
Quadratic Equations ◽  
Positive Distribution ◽  
Langevin Approach ◽  
Basis Method ◽  
Special Case ◽  
Complex Density

The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.

scholarly journals A Multidomain Spectral Method for Analysis of Interior Vibroacoustic Systems with Segmented Boundaries

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Zhenguo Zhang ◽  
Haiting Yu ◽  
Ningyuan Duan ◽  
Hongxing Hua
Keyword(s):  
Finite Element ◽  
Elastic Solid ◽  
Acoustic Medium ◽  
Generalized Variational Principle ◽  
Structural Acoustics ◽  
High Frequencies ◽  
Single Plate ◽  
Structure Assembly ◽  
High Computational Efficiency ◽  
Discretization Procedure

Spectral methods have previously been applied to analyze a multitude of vibration and acoustic problems due to their high computational efficiency. However, their application to interior structural acoustics systems has been limited to the analysis of a single plate coupled to a fluid-filled cavity. In this work, a general multidomain spectral approach is proposed for the eigenvalue and steady-state vibroacoustic analyses of interior structural-acoustic problems with discontinuous boundaries. The unified formulation is derived by means of a generalized variational principle in conjunction with the spectral discretization procedure. The established framework enables one to easily accommodate complex systems consisting of both a structure assembly and a built-up cavity with moderate geometric complexities and to effectively analyze vibroacoustic behaviors with sufficient accuracy at relatively high frequencies. Two practical examples are chosen to demonstrate the flexibility and efficiency of the proposed formulation: a built-up cavity backed by an assembly of multiple connected plates with arbitrary orientations and a thick irregular elastic solid coupled with a heavy acoustic medium. Comparison to finite element simulations and convergence studies for these two examples illustrate the considerable computational advantage of the method as compared to finite element procedures.

THE EQUATION OF MOTION OF A GEOPHONE ON THE SURFACE OF AN ELASTIC EARTH

Geophysics ◽  
1944 ◽  
Vol 9 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Alfred Wolf
Keyword(s):  
Wave Length ◽  
Elastic Solid ◽  
Damping Force ◽  
High Frequencies ◽  
Elastic Earth ◽  
Linear Dimensions ◽  
The Earth ◽  
Restoring Forces ◽  
System Equations

The motion of a geophone case placed on the surface of an elastic earth does not follow faithfully the motion of the earth at high frequencies. In effect, a weight placed on the surface of an elastic solid constitutes a damped oscillating system. The elastic restoring forces are determined by the area of contact between the weight and the surface of the solid and by the elastic moduli of the solid. The damping force is due to emission of elastic waves by the oscillating weight. The motion of the solid also contributes to the inertia of the system. Equations are developed for these forces on the assumption that the wave length is long compared to the linear dimensions of the area of contact between the weight and the elastic solid. This leads to a determination of the frequency of oscillation and of the decrement of such a system.

A theory of compressional wave attenuation in noncohesive sediments

Geophysics ◽  
1985 ◽  
Vol 50 (8) ◽  
pp. 1311-1317 ◽  
Author(s):  
C. McCann ◽  
D. M. McCann
Keyword(s):  
Wave Attenuation ◽  
Compressional Wave ◽  
Solid Particles ◽  
Biot’S Theory ◽  
Attenuation Coefficients ◽  
Low Frequencies ◽  
Biot's Theory ◽  
Compressional Waves ◽  
High Frequencies ◽  
Water Saturated

Published reviews indicate that attenuation coefficients of compressional waves in noncohesive, water‐saturated sediments vary linearly with frequency. Biot’s theory, which accounts for attenuation in terms of the viscous interaction between the solid particles and pore fluid, predicts in its presently published form variation proportional to [Formula: see text] at low frequencies and [Formula: see text] at high frequencies. A modification of Biot’s theory which incorporates a distribution of pore sizes is presented and shown to give excellent agreement with new and published attenuation data in the frequency range 10 kHz to 2.25 MHz. In particular, a linear variation of attenuation with frequency is predicted in that range.

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